We start by examining the bulk behavior of both measures. As the argument is almost identical to that of [1]} or of [2]}, we only sketch it. As \(\theta \rightarrow \infty \) , we scale the entries \(a, b\)  in the double contour integral representation of the kernel \(K_{sp/o}(a, b)\)  linearly:
\( a \approx \alpha \theta + \mathsf {a}, \qquad b \approx \alpha \theta + \mathsf {b}, \qquad \mathsf {a}, \mathsf {b}\in \mathbb {Z}, \ \alpha \in \mathbb {R}\text{ fixed}.\) 
